3D incompressible flows with small viscosity around distant obstacles

Abstract

In this paper, we analyze the behavior of three-dimensional incompressible flows, with small viscosities $\nu >0$, in the exterior of material obstacles $\Omega _{R} = \Omega _{0} + (R,0,0)$, where $\Omega _{0}$ belongs to a class of smooth bounded domains and $R>0$ is sufficiently large. Applying techniques developed by Kato, we prove an explicit energy estimate which, in particular, indicates the limiting flow, when both $\nu \to 0$ and $R\to \infty$, as that one governed by the Euler equations in the whole space. According to this approach, it is natural to contrast our main result to that one already known in the literature for families of viscous flows in expanding domains.

Keywords

• singular perturbation in context of pdes
• vanishing viscosity limit
• navier–stokes equations
• euler equations